Since our other two methods are effectively adaptive (although the adaptivity
is implemented differently), we use an adaptive-bandwidth Gaussian kernel
to calculate the density. To describe it as simply and transparently
as possible we first explain the 1D univariate case and then 3D.
In 1D one first starts by estimating the density with a fixed
bandwidth (\(h\) ) where the Gaussian kernel (\(K\) ) is given
by Equation REF . Equation REF  is then the density estimate
(\(p\) ) for the 1-D fixed bandwidth case where the points are given by \(x_{i}\) .
To estimate the variable or adaptive 1D KDE one allows
the bandwidth to vary from point to point. Let \(d_{i,j}\)  represent
the distance from point \(x_{i}\)  to the kth nearest point in the set
making up the other \(n-1\)  data points. Equation REF 
represents the 1D variable KDE where one sees that the window width
of the kernel at point \(x_{i}\)  is proportional to \(d_{i,j}\)  such
that regions with sparser data points will have flatter kernels.
Hence the new adaptive bandwidth could be represented
as \(h_{i}=h \times d_{i,j}\) . This estimation method is based on
the approach laid out by [1]}.
